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Lattice QCD study for relation between confinement and chiral symmetry breaking on temporally odd-number lattice

We investigate the contribution from each Dirac modes to the Polyakov loop based on a gauge-invariant analytical relation connecting the Polyakov loop and the Dirac modes on a temporally odd-number lattice, where the temporal lattice size is odd, with the normal (nontwisted) periodic boundary condition. The dumping factor in the relation plays crucial role for the negligible contribution of low-lying Dirac modes to the Polyakov loop. The zero-value of the Polyakov loop in the confinement phase is due to the ``positive/negative symmetry'' of the Dirac-mode contribution to the Polyakov loop. In the deconfinement phase, there is no such symmetry.